MidtermSolutions Winter07

Structural Analysis (6th Edition)

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Unformatted text preview: UNIVERSITY OF MICHIGAN Official Examination Booklet Name of Student: Date of Examination: March 9, 2007 Course:- CEE212 — Solid and Structured Mechanics NOTEPLEASEHANDINYOURFORMULASHEETWITHEXAM The University of Michigan Honor Code The Honor Code outlines standards for ethical conduct for graduate and undergraduate students, faculty members, and administrators of the College of Engineering at the University of Michigan. Policies When Taking Exam The instructor need not monitor the exam The instructor will announce the time of the exam and the instructor‘s whereabouts will be communicated to the ciass Students will allow at least one empty seat between themselves and their neighbor The instructor will inform the class prior to the exam if aids are allowed during the exam Students must write and sign the Honor Pledge on their exams The Honor Pledge is: I acknowledge that I have neither given nor received aid on this examination nor have I concealed any violation of the Honor Code. (Signed) QUESTION #1 (25 POINTS) — DESIGN OF AN OVERHANGING STEEL BEAM ,mrtwitttti i: W L t em» re A steel beam is simply supported along its length, as shown above. The beam is overhanging its pin support by 150 mm. The loaded beam is loaded with a uniform intensity of 3.5 kN/m over the full length. The beam cross section is rectangular with a height-width ratio 2 to l (as shown). However, the width of the beam (19) is not specified. The allowable bending stress in the steel is 60 MPa. a) Draw a shear and moment diagram of the beam. b) What is the width, b, of the beam if the maximum beam stress does not exceed the allowable stress? 0) Draw a side View picture of the strain field within the cross section of the beam at the location of the pin support (150 mm from the right-most end of the beam). Clearly label key values including minimum and maximum strain. d) If the beam is loaded beyond the current load of 3.5 kN/m, what distributed load, w, is needed to force the beam to fail due to a “plastic” hinge forming in the beam. Assume the steel is perfectly eiasto—plastic. : .SKH/m A A! f {j \I ‘ ”MIL" m: «1 7/T\F E» _c< 7W3” Twas , :4 8 , 4"” fiprm‘m" ng - 0-393 $1M \/ M 2 10) MWW: Vii: : 0 03:53ng M .2. .1: : alt/Lumen? / MUt‘I’Sflom 1%? \ M :7 0",: 250 Ma; w; (2/. 8? Wm QUESTION #2 (25 PTS) — FLOOD RESISTANT HOUSING FOR NEW ORLEANS Hurricane Katrina resulted in severe devastation to the City of New Orleans, due in part to excessive flooding in the city’s residential sectors (wards). As a result, the chief city engineer is proposing all future housing built in flood prone regions to be designed on steel piers (piles) driven into the ground. A picture of the engineer’s proposed housing scheme is shown above. The steel pier is impacted into the soil down to a depth of L] with a height of L2 left exposed. The house supported by the pier imparts a load of P onto the top of the pier. In response to the load, the soiE resists with a friction force f per unit length of pier. The intensity of the friction varies iinearly with the distance y from the base of the pier by the linearity constant c (as shown). The pier has a cross—sectional area of A and a modulus of elasticity of E. a) Provide an expression for the linear coefficient of friction, c, in terms of the system properties (6.3. L1, L2, y, P, E, A, etc.) b) Derive a formuia for the shortening, 6, of the pile in terms of the system properties. 0) Draw a diagram detailing how the compressive stress, a, varies as a function of the height of the entire pier. P n ; SW] QQUIHSKWM '. ZF=0 14 “M P- a/ :0 ‘1 Mil 9:0}! 0 7P7 iii La ,afcydyfl )0“ g /D :O P— {if no 2. , a: 2/9 Lf" :ZPE 0’), 1., Tom (I? “No 15%: f6 fl ’12? t/ , 2,9 1 [92,. m ' 2),)" 1% 7/” PL» "$2. flag. ’9 ’ l 2* L, H 5"(‘/;r M) $H0m§wwcz 0‘“ d7! r/J: 1:: W 2541,?” 70m gwozmmm of“ 5mm flmmm ‘- L: , F9 F3 {’4’ /$,2%/ 3%;2 0 Swanmiwa or F266 1%mom’ PLZ S12. T EA PL, 2 354 QUESTION #3 (15 POINTS) — ACADEMIC EXERCISE IN TORQUED PIPES A circular rod AB with fixed ends has a hole extending for half its length as shown below. The diameter of the hole is three fourths of the outer diameter of the rod. At what distanced x from the left-hand end should a torque, T, be applied so that the reactive torques at the suppofis will be equal? fluggwom 33‘: 3 EWILEEKIUM' m REMOVE 13 : [dad firsurte 7'15- Howow F’ormpiu fl)?” SUPPORT 0N '. i: CUr‘FPrflEH/EE ' 4785498 T]: + M ._ ”EL- 4, TEL 261 \EAF 6? 3:53 262324; ”-51.3an TL. __ :f 4‘ 7—;(X L71) 2‘ EL {- 7E}... 4” C J” —-——._m “—5 7 pg 243;? 251013 gm fax X2§~(3-E§> 3% ‘§“ E: M __.= M 4 <3 4:34 QUESTION #4 (15 POINTS) — DESIGN OF A REINFORCED WOOD BEAM In California, residential construction is primarily designed using wood construction. Unfortunateiy, wood is not always the strongest material when loaded in fiexural bending. For California, the issue is more critical due to the occurrence of seismic loading in the region. As a result, you conceive of a means of strengthening wood beams using steel plates. A picture of the proposed section is shown below: Consider the modulus of elasticity for the wood as 12.5 GPa and 200 GPa for the steel plates. If a bending moment of 50 kN-m is applied to the composite beam, determine: a) Maximum stress in the wood b) Stress in the steei along the top edge [w : ”.5634 E3 3 26294 “Oak. TKANSFMfiM Sécmsm m 15.3- 3 /6 EN 9 4) END NavmmL SURFAQE (X) (I g 25:5 59 zng 0.52%4 Z 5 9.32.45 (3.342 0-5'7" + 0.55“)< 3 iii—g (0.47M; # 0,3104 0 1: (15.20}, i X 2.. é Ew>< +0,+ro; g: 0'3”“ z X >< gowafirXiflZW" {fax/o 3) (a, xg) 9?. #5290 ’ 3 a? “*3 457M? wWWWw—m an 53“. ”5:"- {5) “m”? 4515Mqu QUESTEON #5 (20 POINTS) — General Questions of Theoretical Principles l) Please list the three laws upon which all of the principles of solid mechanics are based? .__ EQUILIEEIUM ,. (we SflTWiu a»; .. (bf/{Pameze m/ 2) What assumption must be made before it can be assumed that strain varies linearly over the section of a beam under flexural loading? ”“ P/fima Sac/é'hw Véé/flfiiy} fig/me, 3) What is the definition of the neutral axis? Also, how do you determine where the neutral axis is? #0 Egalesfifiiofl 5 £4in 4) Draw a typical stress—strain diagram for concrete — include both tension and compression properties. (3‘ (Tan S'lDV} 5/. Com Preset 9% 5) When you have a reinforced concrete beam under applied bending (as shown below), draw the strain and stress profile of the beam commonly used to analyze the beam flexural capacity. Strain Stress ...
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